Optimization problems and replica symmetry breaking in finite connectivity spin glasses

A formalism capable of handling the first step of hierarchical replica symmetry breaking (RSB) in finite-connectivity models is introduced. The emerging order parameter is claimed to be a probability distribution over the space of field distributions (or, equivalently magnetization distributions) inside the cluster of states. The approach is shown to coincide with previous works in the replica-symmetric case and in the two limiting cases m = 0 and 1 where m is Parisi's break point. As an application to the study of optimization problems, the GS properties of the random 3-satisfiability problem are investigated and we present a first RSB solution improving replica-symmetric results.

[1]  Monasson,et al.  Entropy of the K-satisfiability problem. , 1996, Physical review letters.

[2]  Paul G. Spirakis,et al.  Tail bounds for occupancy and the satisfiability threshold conjecture , 1994, Proceedings 35th Annual Symposium on Foundations of Computer Science.

[3]  S Kirkpatrick,et al.  Critical Behavior in the Satisfiability of Random Boolean Expressions , 1994, Science.

[4]  Olivier Dubois,et al.  Counting the Number of Solutions for Instances of Satisfiability , 1991, Theor. Comput. Sci..

[5]  Pik-Yin Lai,et al.  The finite connectivity spin glass: investigation of replica symmetry breaking of the ground state , 1990 .

[6]  Huifang,et al.  Griffiths singularities in random magnets: Results for a soluble model. , 1989, Physical review. B, Condensed matter.

[7]  D Sherrington,et al.  Intensively connected spin glasses: towards a replica-symmetry-breaking solution of the ground state , 1988 .

[8]  E. Gardner,et al.  Optimal storage properties of neural network models , 1988 .

[9]  E. Gardner.  The space of interactions in neural network models , 1988 .

[10]  P. Mottishaw,et al.  Replica symmetry breaking in weak connectivity systems , 1987 .

[11]  P. Mottishaw,et al.  On the stability of randomly frustrated systems with finite connectivity , 1987 .

[12]  Kanter,et al.  Mean-field theory of spin-glasses with finite coordination number. , 1987, Physical review letters.

[13]  H. Orland.  Mean-field theory for optimization problems , 1985 .

[14]  A. Bray,et al.  Phase diagrams for dilute spin glasses , 1985 .

[15]  E. Gardner.  Spin glasses with p-spin interactions , 1985 .

[16]  M. Mézard,et al.  The simplest spin glass , 1984 .

[17]  M. Garey.  Johnson: computers and intractability: a guide to the theory of np- completeness (freeman , 1979 .

[18]  David S. Johnson,et al.  Computers and Intractability: A Guide to the Theory of NP-Completeness , 1978 .

[19]  R. Palmer,et al.  Solution of 'Solvable model of a spin glass' , 1977 .

[20]  S. Kirkpatrick,et al.  Solvable Model of a Spin-Glass , 1975 .

[21]  B. Hayes.  The American Scientist , 1962, Nature.